Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C4×#121 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C23 |
| commutator | C22 | C23×C4 |
| Frattini, socle, cosocle | C23 | C24 |
| prime radical | C2×#121 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
412 conjugacy classes of subgroups; only the 172 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 172 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 12 points:
(1,2,3,4);(6,8)(9,10)(11,12);(5,6,7,8);(10,12)56 conjugacy classes — view the full character table.
| maximal subgroup of | #7565, #7582, #7586, … |
| maximal normal subgroup of | #7565, #7582, #7586, … |
| maximal quotient of | C4×#386, C4×#388, C4×#390, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1032);